Mobius ladder

After synthesizing the 3-rung Mobius ladder, Walba conjectured that its molecular bond graph was topologically chiral [16]. In contrast with his conjecture, he... 

Figure 23. A 3-rung and a 4-rung molecular Mobius ladder.

Figure 24. A Mobius ladder with three rungs can be deformed so that it has a planar reflection.

Figure 25. A two-rung Mobius ladder can be deformed into a plane.

Figure 27. A topologically achiral embedding of a Mobius ladder with four rungs.

In Figure 34, we have labeled the vertices of the Mobius ladder with 1, 2, 3 and a, b, c, in order to illustrate that the 3-rung Mobius ladder is actually a K3 3 graph. 

Figure 34. A molecular Mobius ladder with three mngs is an embedding of K3 3.

Figure 38. A 4-rung Mobius ladder contained in the molecular graph of Figure 37.

Figure 39. This rearrangement of the vertices is induced by an automorphism of the Mobius ladder of Figure 38.

On the other hand, there is no automorphism of the Mobius ladder of Figure 38 which could rearrange the vertices as illustrated in Figure 40. This is because the Mobius ladder of Figure 38 has vertex number 3 adjacent to vertex number 7, but the Mobius ladder of Figure 40 does not have these two vertices adjacent. 

Figure 16. Some topologically chiral molecular graphs, (a) The Simmons-Paquette molecule.74 (b) Walba s 3-rung Mobius ladder molecule.75 (c) [4](l,l )[4](3,3 )[3](4 4 )-Ferrocenophan-16-one.76 (d) Triple-layered naphthalenophane.78 Unlabeled vertices represent carbon atoms, and hydrogen atoms are suppressed for clarity. Only one enantiomer of each molecule is shown.

The first flexible, topologically chiral K3j3 molecule is the so-called three-rung Mobius ladder 63 (Fig. 15). It was synthesized by D. M. Walba et al. in 1982 [85] and is represented on Fig. 15, together with several presentations of its graph (64-66). It is important to notice that the rungs of the ladder are different from... 

Fig. 15. The topologically chiral three-rung Mobius ladder (63) and several presentations of its graph (64-66). 64 and 66 are presentations with the minimum number of crossings (1). Note that the edges (1,1 ), (2,2 ) and (3,3 ) are differently coloured than the others (i.e. (l, 2), (2,3 ), etc,. ..)...

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